Abstract

The enhancement of second-harmonic generation (SHG) across a 2D photonic crystal (PC) slab consisting of GaP is investigated in this study with the three-dimensional spectral element method (SEM). The in-plane band structure is calculated, and it is compared with the peaks of the SHG to reveal the mechanisms behind the enhancement. The numerical result from the SEM shows that, under normal incidence, the scattered power of the SHG is enhanced for the eigenstates with large decay rates, while the stored energy of the SHG is enhanced for the eigenstates with a zero decay rate. The SHG is enhanced under two conditions: (i) phase matching between the fundamental and second-harmonic (SH) fields and (ii) symmetry matching between the field pattern of the resonant eigenstate and the generated SH polarization field. Compared with a homogeneous dielectric slab, the air-bridge PC slab can enhance the SHG by 4 orders of magnitude.

© 2011 Optical Society of America

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2011 (1)

2010 (1)

R. Iliew, C. Etrich, T. Pertsch, F. Lederer, and Y. S. Kivshar, “Huge enhancement of backward second-harmonic generation with slow light in photonic crystals,” Phys. Rev. A 81, 023820(2010).
[CrossRef]

2009 (1)

M. Luo and Q. H. Liu, “Spectral element method for band structures of three-dimensional anisotropic photonic crystals,” Phys. Rev. E 80, 056702 (2009).
[CrossRef]

2008 (1)

B. Maes, P. Bienstman, R. Baets, B. Hu, P. Sewell, and T. Benson, “Modeling comparison of second-harmonic generation in high-index-contrast devices,” Opt. Quantum Electron. 40, 13-22(2008).
[CrossRef]

2007 (2)

D. C. Marinica, A. G. Borisov, and S. V. Shabanov, “Second harmonic generation from arrays of subwavelength cylinders,” Phys. Rev. B 76, 085311 (2007).
[CrossRef]

J. Bravo-Abad, A. Rodriguez, P. Bermel, S. G. Johnson, J. D. Joannopoulos, and M. Soljacic, “Enhanced nonlinear optics in photonic-crystal microcavities,” Opt. Express 15, 16161-16176(2007).
[CrossRef] [PubMed]

2006 (2)

M. Liscidini and L. C. Andreani, “Second-harmonic generation in doubly resonant microcavities with periodic dielectric mirrors,” Phys. Rev. E 73, 016613 (2006).
[CrossRef]

K. B. Crozier, V. Lousse, O. Kilic, S. Kim, S. Fan, and O. Solgaard, “Air-bridged photonic crystal slabs at visible and near-infrared wavelengths,” Phys. Rev. B 73, 115126 (2006).
[CrossRef]

2004 (2)

M. Soljacic and J. D. Joannopoulos, “Enhancement of nonlinear effects using photonic crystals,” Nat. Mater. 3, 211-219 (2004).
[CrossRef] [PubMed]

J. Torres, D. Coquillat, R. Legros, J. P. Lascaray, F. Teppe, D. Scalbert, D. Peyrade, Y. Chen, O. Briot, M. Le Vassor d'Yerville, E. Centeno, D. Cassagne, and J. P. Albert, “Giant second-harmonic generation in a one-dimensional GaN photonic crystal,” Phys. Rev. B 69, 085105 (2004).
[CrossRef]

2003 (2)

Y. Dumeige, F. Raineri, A. Levenson, and X. Letartre, “Second-harmonic generation in one-dimensional photonic edge waveguides,” Phys. Rev. B 68, 066617 (2003).
[CrossRef]

A. M. Malvezzi, G. Vecchi, M. Patrini, G. Guizzetti, L. C. Andreani, F. Romanato, L. Businaro, E. Di Fabrizio, A. Passaseo, and M. D. Vittorio, “Resonant second-harmonic generation in a GaAs photonic crystal waveguide,” Phys. Rev. B 68, 161306(R) (2003).
[CrossRef]

2002 (2)

A. R. Cowan and J. F. Young, “Mode matching for second-harmonic generation in photonic crystal waveguides,” Phys. Rev. B 65, 085106 (2002).
[CrossRef]

Y. Dumeige, I. Sagnes, P. Monnier, P. Vidakovic, I. Abram, C. Mriadec, and A. Levenson, “Phase-matched frequency doubling at photonic band edges: efficiency scaling as the fifth power of the length,” Phys. Rev. Lett. 89, 043901 (2002).
[CrossRef] [PubMed]

1999 (1)

M. Centini, C. Sibilia, M. Scalora, G. D'Aguanno, M. Bertolotti, M. J. Bloemer, C. M. Bowden, and I. Nefedov, “Dispersive properties of finite, one-dimensional photonic band gap structures: applications to nonlinear quadratic interactions,” Phys. Rev. E 60, 4891-4898 (1999).
[CrossRef]

1997 (1)

1996 (1)

J. S. Savage and A. F. Peterson, “Higher-order vector finite elements for tetrahedral cells,” IEEE Trans. Microw. Theory Technol. 44, 874-879 (1996).
[CrossRef]

1983 (1)

D. E. Aspnes and A. A. Studna, “Dielectric functions and optical parameters of Si, Ge, GaP, GaAs, GaSb, InP, InAs, and InSb from 1.5 to 6.0 eV,” Phys. Rev. B 27, 985-1009 (1983).
[CrossRef]

1966 (1)

S. E. Harris, “Proposed backward wave oscillation in the infrared,” Appl. Phys. Lett. 9, 114 (1966).
[CrossRef]

1961 (1)

P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118-119(1961).
[CrossRef]

Abram, I.

Y. Dumeige, I. Sagnes, P. Monnier, P. Vidakovic, I. Abram, C. Mriadec, and A. Levenson, “Phase-matched frequency doubling at photonic band edges: efficiency scaling as the fifth power of the length,” Phys. Rev. Lett. 89, 043901 (2002).
[CrossRef] [PubMed]

Albert, J. P.

J. Torres, D. Coquillat, R. Legros, J. P. Lascaray, F. Teppe, D. Scalbert, D. Peyrade, Y. Chen, O. Briot, M. Le Vassor d'Yerville, E. Centeno, D. Cassagne, and J. P. Albert, “Giant second-harmonic generation in a one-dimensional GaN photonic crystal,” Phys. Rev. B 69, 085105 (2004).
[CrossRef]

Andreani, L. C.

M. Liscidini and L. C. Andreani, “Second-harmonic generation in doubly resonant microcavities with periodic dielectric mirrors,” Phys. Rev. E 73, 016613 (2006).
[CrossRef]

A. M. Malvezzi, G. Vecchi, M. Patrini, G. Guizzetti, L. C. Andreani, F. Romanato, L. Businaro, E. Di Fabrizio, A. Passaseo, and M. D. Vittorio, “Resonant second-harmonic generation in a GaAs photonic crystal waveguide,” Phys. Rev. B 68, 161306(R) (2003).
[CrossRef]

Aspnes, D. E.

D. E. Aspnes and A. A. Studna, “Dielectric functions and optical parameters of Si, Ge, GaP, GaAs, GaSb, InP, InAs, and InSb from 1.5 to 6.0 eV,” Phys. Rev. B 27, 985-1009 (1983).
[CrossRef]

Baets, R.

B. Maes, P. Bienstman, R. Baets, B. Hu, P. Sewell, and T. Benson, “Modeling comparison of second-harmonic generation in high-index-contrast devices,” Opt. Quantum Electron. 40, 13-22(2008).
[CrossRef]

Benson, T.

B. Maes, P. Bienstman, R. Baets, B. Hu, P. Sewell, and T. Benson, “Modeling comparison of second-harmonic generation in high-index-contrast devices,” Opt. Quantum Electron. 40, 13-22(2008).
[CrossRef]

Bermel, P.

Bertolotti, M.

M. Centini, C. Sibilia, M. Scalora, G. D'Aguanno, M. Bertolotti, M. J. Bloemer, C. M. Bowden, and I. Nefedov, “Dispersive properties of finite, one-dimensional photonic band gap structures: applications to nonlinear quadratic interactions,” Phys. Rev. E 60, 4891-4898 (1999).
[CrossRef]

Bienstman, P.

B. Maes, P. Bienstman, R. Baets, B. Hu, P. Sewell, and T. Benson, “Modeling comparison of second-harmonic generation in high-index-contrast devices,” Opt. Quantum Electron. 40, 13-22(2008).
[CrossRef]

Bloemer, M. J.

M. Centini, C. Sibilia, M. Scalora, G. D'Aguanno, M. Bertolotti, M. J. Bloemer, C. M. Bowden, and I. Nefedov, “Dispersive properties of finite, one-dimensional photonic band gap structures: applications to nonlinear quadratic interactions,” Phys. Rev. E 60, 4891-4898 (1999).
[CrossRef]

Borisov, A. G.

D. C. Marinica, A. G. Borisov, and S. V. Shabanov, “Second harmonic generation from arrays of subwavelength cylinders,” Phys. Rev. B 76, 085311 (2007).
[CrossRef]

Bowden, C. M.

M. Centini, C. Sibilia, M. Scalora, G. D'Aguanno, M. Bertolotti, M. J. Bloemer, C. M. Bowden, and I. Nefedov, “Dispersive properties of finite, one-dimensional photonic band gap structures: applications to nonlinear quadratic interactions,” Phys. Rev. E 60, 4891-4898 (1999).
[CrossRef]

Bravo-Abad, J.

Briot, O.

J. Torres, D. Coquillat, R. Legros, J. P. Lascaray, F. Teppe, D. Scalbert, D. Peyrade, Y. Chen, O. Briot, M. Le Vassor d'Yerville, E. Centeno, D. Cassagne, and J. P. Albert, “Giant second-harmonic generation in a one-dimensional GaN photonic crystal,” Phys. Rev. B 69, 085105 (2004).
[CrossRef]

Businaro, L.

A. M. Malvezzi, G. Vecchi, M. Patrini, G. Guizzetti, L. C. Andreani, F. Romanato, L. Businaro, E. Di Fabrizio, A. Passaseo, and M. D. Vittorio, “Resonant second-harmonic generation in a GaAs photonic crystal waveguide,” Phys. Rev. B 68, 161306(R) (2003).
[CrossRef]

Cassagne, D.

J. Torres, D. Coquillat, R. Legros, J. P. Lascaray, F. Teppe, D. Scalbert, D. Peyrade, Y. Chen, O. Briot, M. Le Vassor d'Yerville, E. Centeno, D. Cassagne, and J. P. Albert, “Giant second-harmonic generation in a one-dimensional GaN photonic crystal,” Phys. Rev. B 69, 085105 (2004).
[CrossRef]

Centeno, E.

J. Torres, D. Coquillat, R. Legros, J. P. Lascaray, F. Teppe, D. Scalbert, D. Peyrade, Y. Chen, O. Briot, M. Le Vassor d'Yerville, E. Centeno, D. Cassagne, and J. P. Albert, “Giant second-harmonic generation in a one-dimensional GaN photonic crystal,” Phys. Rev. B 69, 085105 (2004).
[CrossRef]

Centini, M.

M. Centini, C. Sibilia, M. Scalora, G. D'Aguanno, M. Bertolotti, M. J. Bloemer, C. M. Bowden, and I. Nefedov, “Dispersive properties of finite, one-dimensional photonic band gap structures: applications to nonlinear quadratic interactions,” Phys. Rev. E 60, 4891-4898 (1999).
[CrossRef]

Chen, Y.

J. Torres, D. Coquillat, R. Legros, J. P. Lascaray, F. Teppe, D. Scalbert, D. Peyrade, Y. Chen, O. Briot, M. Le Vassor d'Yerville, E. Centeno, D. Cassagne, and J. P. Albert, “Giant second-harmonic generation in a one-dimensional GaN photonic crystal,” Phys. Rev. B 69, 085105 (2004).
[CrossRef]

Coquillat, D.

J. Torres, D. Coquillat, R. Legros, J. P. Lascaray, F. Teppe, D. Scalbert, D. Peyrade, Y. Chen, O. Briot, M. Le Vassor d'Yerville, E. Centeno, D. Cassagne, and J. P. Albert, “Giant second-harmonic generation in a one-dimensional GaN photonic crystal,” Phys. Rev. B 69, 085105 (2004).
[CrossRef]

Cowan, A. R.

A. R. Cowan and J. F. Young, “Mode matching for second-harmonic generation in photonic crystal waveguides,” Phys. Rev. B 65, 085106 (2002).
[CrossRef]

Crozier, K. B.

K. B. Crozier, V. Lousse, O. Kilic, S. Kim, S. Fan, and O. Solgaard, “Air-bridged photonic crystal slabs at visible and near-infrared wavelengths,” Phys. Rev. B 73, 115126 (2006).
[CrossRef]

D'Aguanno, G.

M. Centini, C. Sibilia, M. Scalora, G. D'Aguanno, M. Bertolotti, M. J. Bloemer, C. M. Bowden, and I. Nefedov, “Dispersive properties of finite, one-dimensional photonic band gap structures: applications to nonlinear quadratic interactions,” Phys. Rev. E 60, 4891-4898 (1999).
[CrossRef]

Di Fabrizio, E.

A. M. Malvezzi, G. Vecchi, M. Patrini, G. Guizzetti, L. C. Andreani, F. Romanato, L. Businaro, E. Di Fabrizio, A. Passaseo, and M. D. Vittorio, “Resonant second-harmonic generation in a GaAs photonic crystal waveguide,” Phys. Rev. B 68, 161306(R) (2003).
[CrossRef]

Dumeige, Y.

Y. Dumeige, F. Raineri, A. Levenson, and X. Letartre, “Second-harmonic generation in one-dimensional photonic edge waveguides,” Phys. Rev. B 68, 066617 (2003).
[CrossRef]

Y. Dumeige, I. Sagnes, P. Monnier, P. Vidakovic, I. Abram, C. Mriadec, and A. Levenson, “Phase-matched frequency doubling at photonic band edges: efficiency scaling as the fifth power of the length,” Phys. Rev. Lett. 89, 043901 (2002).
[CrossRef] [PubMed]

Etrich, C.

R. Iliew, C. Etrich, T. Pertsch, F. Lederer, and Y. S. Kivshar, “Huge enhancement of backward second-harmonic generation with slow light in photonic crystals,” Phys. Rev. A 81, 023820(2010).
[CrossRef]

Fan, S.

K. B. Crozier, V. Lousse, O. Kilic, S. Kim, S. Fan, and O. Solgaard, “Air-bridged photonic crystal slabs at visible and near-infrared wavelengths,” Phys. Rev. B 73, 115126 (2006).
[CrossRef]

Franken, P. A.

P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118-119(1961).
[CrossRef]

Guizzetti, G.

A. M. Malvezzi, G. Vecchi, M. Patrini, G. Guizzetti, L. C. Andreani, F. Romanato, L. Businaro, E. Di Fabrizio, A. Passaseo, and M. D. Vittorio, “Resonant second-harmonic generation in a GaAs photonic crystal waveguide,” Phys. Rev. B 68, 161306(R) (2003).
[CrossRef]

Harris, S. E.

S. E. Harris, “Proposed backward wave oscillation in the infrared,” Appl. Phys. Lett. 9, 114 (1966).
[CrossRef]

Hill, A. E.

P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118-119(1961).
[CrossRef]

Hu, B.

B. Maes, P. Bienstman, R. Baets, B. Hu, P. Sewell, and T. Benson, “Modeling comparison of second-harmonic generation in high-index-contrast devices,” Opt. Quantum Electron. 40, 13-22(2008).
[CrossRef]

Iliew, R.

R. Iliew, C. Etrich, T. Pertsch, F. Lederer, and Y. S. Kivshar, “Huge enhancement of backward second-harmonic generation with slow light in photonic crystals,” Phys. Rev. A 81, 023820(2010).
[CrossRef]

Ito, R.

Joannopoulos, J. D.

Johnson, S. G.

Kilic, O.

K. B. Crozier, V. Lousse, O. Kilic, S. Kim, S. Fan, and O. Solgaard, “Air-bridged photonic crystal slabs at visible and near-infrared wavelengths,” Phys. Rev. B 73, 115126 (2006).
[CrossRef]

Kim, S.

K. B. Crozier, V. Lousse, O. Kilic, S. Kim, S. Fan, and O. Solgaard, “Air-bridged photonic crystal slabs at visible and near-infrared wavelengths,” Phys. Rev. B 73, 115126 (2006).
[CrossRef]

Kitamoto, A.

Kivshar, Y. S.

R. Iliew, C. Etrich, T. Pertsch, F. Lederer, and Y. S. Kivshar, “Huge enhancement of backward second-harmonic generation with slow light in photonic crystals,” Phys. Rev. A 81, 023820(2010).
[CrossRef]

Kondo, T.

Lascaray, J. P.

J. Torres, D. Coquillat, R. Legros, J. P. Lascaray, F. Teppe, D. Scalbert, D. Peyrade, Y. Chen, O. Briot, M. Le Vassor d'Yerville, E. Centeno, D. Cassagne, and J. P. Albert, “Giant second-harmonic generation in a one-dimensional GaN photonic crystal,” Phys. Rev. B 69, 085105 (2004).
[CrossRef]

Le Vassor d'Yerville, M.

J. Torres, D. Coquillat, R. Legros, J. P. Lascaray, F. Teppe, D. Scalbert, D. Peyrade, Y. Chen, O. Briot, M. Le Vassor d'Yerville, E. Centeno, D. Cassagne, and J. P. Albert, “Giant second-harmonic generation in a one-dimensional GaN photonic crystal,” Phys. Rev. B 69, 085105 (2004).
[CrossRef]

Lederer, F.

R. Iliew, C. Etrich, T. Pertsch, F. Lederer, and Y. S. Kivshar, “Huge enhancement of backward second-harmonic generation with slow light in photonic crystals,” Phys. Rev. A 81, 023820(2010).
[CrossRef]

Legros, R.

J. Torres, D. Coquillat, R. Legros, J. P. Lascaray, F. Teppe, D. Scalbert, D. Peyrade, Y. Chen, O. Briot, M. Le Vassor d'Yerville, E. Centeno, D. Cassagne, and J. P. Albert, “Giant second-harmonic generation in a one-dimensional GaN photonic crystal,” Phys. Rev. B 69, 085105 (2004).
[CrossRef]

Letartre, X.

Y. Dumeige, F. Raineri, A. Levenson, and X. Letartre, “Second-harmonic generation in one-dimensional photonic edge waveguides,” Phys. Rev. B 68, 066617 (2003).
[CrossRef]

Levenson, A.

Y. Dumeige, F. Raineri, A. Levenson, and X. Letartre, “Second-harmonic generation in one-dimensional photonic edge waveguides,” Phys. Rev. B 68, 066617 (2003).
[CrossRef]

Y. Dumeige, I. Sagnes, P. Monnier, P. Vidakovic, I. Abram, C. Mriadec, and A. Levenson, “Phase-matched frequency doubling at photonic band edges: efficiency scaling as the fifth power of the length,” Phys. Rev. Lett. 89, 043901 (2002).
[CrossRef] [PubMed]

Liscidini, M.

M. Liscidini and L. C. Andreani, “Second-harmonic generation in doubly resonant microcavities with periodic dielectric mirrors,” Phys. Rev. E 73, 016613 (2006).
[CrossRef]

Liu, Q. H.

M. Luo and Q. H. Liu, “Extraordinary transmission of a thick film with a periodic structure consisting of strongly dispersive materials,” J. Opt. Soc. Am. B 28, 629-636 (2011).
[CrossRef]

M. Luo and Q. H. Liu, “Spectral element method for band structures of three-dimensional anisotropic photonic crystals,” Phys. Rev. E 80, 056702 (2009).
[CrossRef]

Lousse, V.

K. B. Crozier, V. Lousse, O. Kilic, S. Kim, S. Fan, and O. Solgaard, “Air-bridged photonic crystal slabs at visible and near-infrared wavelengths,” Phys. Rev. B 73, 115126 (2006).
[CrossRef]

Luo, M.

M. Luo and Q. H. Liu, “Extraordinary transmission of a thick film with a periodic structure consisting of strongly dispersive materials,” J. Opt. Soc. Am. B 28, 629-636 (2011).
[CrossRef]

M. Luo and Q. H. Liu, “Spectral element method for band structures of three-dimensional anisotropic photonic crystals,” Phys. Rev. E 80, 056702 (2009).
[CrossRef]

Maes, B.

B. Maes, P. Bienstman, R. Baets, B. Hu, P. Sewell, and T. Benson, “Modeling comparison of second-harmonic generation in high-index-contrast devices,” Opt. Quantum Electron. 40, 13-22(2008).
[CrossRef]

Malvezzi, A. M.

A. M. Malvezzi, G. Vecchi, M. Patrini, G. Guizzetti, L. C. Andreani, F. Romanato, L. Businaro, E. Di Fabrizio, A. Passaseo, and M. D. Vittorio, “Resonant second-harmonic generation in a GaAs photonic crystal waveguide,” Phys. Rev. B 68, 161306(R) (2003).
[CrossRef]

Marinica, D. C.

D. C. Marinica, A. G. Borisov, and S. V. Shabanov, “Second harmonic generation from arrays of subwavelength cylinders,” Phys. Rev. B 76, 085311 (2007).
[CrossRef]

Monnier, P.

Y. Dumeige, I. Sagnes, P. Monnier, P. Vidakovic, I. Abram, C. Mriadec, and A. Levenson, “Phase-matched frequency doubling at photonic band edges: efficiency scaling as the fifth power of the length,” Phys. Rev. Lett. 89, 043901 (2002).
[CrossRef] [PubMed]

Mriadec, C.

Y. Dumeige, I. Sagnes, P. Monnier, P. Vidakovic, I. Abram, C. Mriadec, and A. Levenson, “Phase-matched frequency doubling at photonic band edges: efficiency scaling as the fifth power of the length,” Phys. Rev. Lett. 89, 043901 (2002).
[CrossRef] [PubMed]

Nefedov, I.

M. Centini, C. Sibilia, M. Scalora, G. D'Aguanno, M. Bertolotti, M. J. Bloemer, C. M. Bowden, and I. Nefedov, “Dispersive properties of finite, one-dimensional photonic band gap structures: applications to nonlinear quadratic interactions,” Phys. Rev. E 60, 4891-4898 (1999).
[CrossRef]

Passaseo, A.

A. M. Malvezzi, G. Vecchi, M. Patrini, G. Guizzetti, L. C. Andreani, F. Romanato, L. Businaro, E. Di Fabrizio, A. Passaseo, and M. D. Vittorio, “Resonant second-harmonic generation in a GaAs photonic crystal waveguide,” Phys. Rev. B 68, 161306(R) (2003).
[CrossRef]

Patrini, M.

A. M. Malvezzi, G. Vecchi, M. Patrini, G. Guizzetti, L. C. Andreani, F. Romanato, L. Businaro, E. Di Fabrizio, A. Passaseo, and M. D. Vittorio, “Resonant second-harmonic generation in a GaAs photonic crystal waveguide,” Phys. Rev. B 68, 161306(R) (2003).
[CrossRef]

Pertsch, T.

R. Iliew, C. Etrich, T. Pertsch, F. Lederer, and Y. S. Kivshar, “Huge enhancement of backward second-harmonic generation with slow light in photonic crystals,” Phys. Rev. A 81, 023820(2010).
[CrossRef]

Peters, C. W.

P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118-119(1961).
[CrossRef]

Peterson, A. F.

J. S. Savage and A. F. Peterson, “Higher-order vector finite elements for tetrahedral cells,” IEEE Trans. Microw. Theory Technol. 44, 874-879 (1996).
[CrossRef]

Peyrade, D.

J. Torres, D. Coquillat, R. Legros, J. P. Lascaray, F. Teppe, D. Scalbert, D. Peyrade, Y. Chen, O. Briot, M. Le Vassor d'Yerville, E. Centeno, D. Cassagne, and J. P. Albert, “Giant second-harmonic generation in a one-dimensional GaN photonic crystal,” Phys. Rev. B 69, 085105 (2004).
[CrossRef]

Raineri, F.

Y. Dumeige, F. Raineri, A. Levenson, and X. Letartre, “Second-harmonic generation in one-dimensional photonic edge waveguides,” Phys. Rev. B 68, 066617 (2003).
[CrossRef]

Rodriguez, A.

Romanato, F.

A. M. Malvezzi, G. Vecchi, M. Patrini, G. Guizzetti, L. C. Andreani, F. Romanato, L. Businaro, E. Di Fabrizio, A. Passaseo, and M. D. Vittorio, “Resonant second-harmonic generation in a GaAs photonic crystal waveguide,” Phys. Rev. B 68, 161306(R) (2003).
[CrossRef]

Sagnes, I.

Y. Dumeige, I. Sagnes, P. Monnier, P. Vidakovic, I. Abram, C. Mriadec, and A. Levenson, “Phase-matched frequency doubling at photonic band edges: efficiency scaling as the fifth power of the length,” Phys. Rev. Lett. 89, 043901 (2002).
[CrossRef] [PubMed]

Savage, J. S.

J. S. Savage and A. F. Peterson, “Higher-order vector finite elements for tetrahedral cells,” IEEE Trans. Microw. Theory Technol. 44, 874-879 (1996).
[CrossRef]

Scalbert, D.

J. Torres, D. Coquillat, R. Legros, J. P. Lascaray, F. Teppe, D. Scalbert, D. Peyrade, Y. Chen, O. Briot, M. Le Vassor d'Yerville, E. Centeno, D. Cassagne, and J. P. Albert, “Giant second-harmonic generation in a one-dimensional GaN photonic crystal,” Phys. Rev. B 69, 085105 (2004).
[CrossRef]

Scalora, M.

M. Centini, C. Sibilia, M. Scalora, G. D'Aguanno, M. Bertolotti, M. J. Bloemer, C. M. Bowden, and I. Nefedov, “Dispersive properties of finite, one-dimensional photonic band gap structures: applications to nonlinear quadratic interactions,” Phys. Rev. E 60, 4891-4898 (1999).
[CrossRef]

Sewell, P.

B. Maes, P. Bienstman, R. Baets, B. Hu, P. Sewell, and T. Benson, “Modeling comparison of second-harmonic generation in high-index-contrast devices,” Opt. Quantum Electron. 40, 13-22(2008).
[CrossRef]

Shabanov, S. V.

D. C. Marinica, A. G. Borisov, and S. V. Shabanov, “Second harmonic generation from arrays of subwavelength cylinders,” Phys. Rev. B 76, 085311 (2007).
[CrossRef]

Shirane, M.

Shoji, I.

Sibilia, C.

M. Centini, C. Sibilia, M. Scalora, G. D'Aguanno, M. Bertolotti, M. J. Bloemer, C. M. Bowden, and I. Nefedov, “Dispersive properties of finite, one-dimensional photonic band gap structures: applications to nonlinear quadratic interactions,” Phys. Rev. E 60, 4891-4898 (1999).
[CrossRef]

Solgaard, O.

K. B. Crozier, V. Lousse, O. Kilic, S. Kim, S. Fan, and O. Solgaard, “Air-bridged photonic crystal slabs at visible and near-infrared wavelengths,” Phys. Rev. B 73, 115126 (2006).
[CrossRef]

Soljacic, M.

Studna, A. A.

D. E. Aspnes and A. A. Studna, “Dielectric functions and optical parameters of Si, Ge, GaP, GaAs, GaSb, InP, InAs, and InSb from 1.5 to 6.0 eV,” Phys. Rev. B 27, 985-1009 (1983).
[CrossRef]

Sutherland, R. L.

R. L. Sutherland, Handbook of Nonlinear Optics, 2nd ed.(Marcel Dekker, 2003).
[CrossRef]

Teppe, F.

J. Torres, D. Coquillat, R. Legros, J. P. Lascaray, F. Teppe, D. Scalbert, D. Peyrade, Y. Chen, O. Briot, M. Le Vassor d'Yerville, E. Centeno, D. Cassagne, and J. P. Albert, “Giant second-harmonic generation in a one-dimensional GaN photonic crystal,” Phys. Rev. B 69, 085105 (2004).
[CrossRef]

Torres, J.

J. Torres, D. Coquillat, R. Legros, J. P. Lascaray, F. Teppe, D. Scalbert, D. Peyrade, Y. Chen, O. Briot, M. Le Vassor d'Yerville, E. Centeno, D. Cassagne, and J. P. Albert, “Giant second-harmonic generation in a one-dimensional GaN photonic crystal,” Phys. Rev. B 69, 085105 (2004).
[CrossRef]

Vecchi, G.

A. M. Malvezzi, G. Vecchi, M. Patrini, G. Guizzetti, L. C. Andreani, F. Romanato, L. Businaro, E. Di Fabrizio, A. Passaseo, and M. D. Vittorio, “Resonant second-harmonic generation in a GaAs photonic crystal waveguide,” Phys. Rev. B 68, 161306(R) (2003).
[CrossRef]

Vidakovic, P.

Y. Dumeige, I. Sagnes, P. Monnier, P. Vidakovic, I. Abram, C. Mriadec, and A. Levenson, “Phase-matched frequency doubling at photonic band edges: efficiency scaling as the fifth power of the length,” Phys. Rev. Lett. 89, 043901 (2002).
[CrossRef] [PubMed]

Vittorio, M. D.

A. M. Malvezzi, G. Vecchi, M. Patrini, G. Guizzetti, L. C. Andreani, F. Romanato, L. Businaro, E. Di Fabrizio, A. Passaseo, and M. D. Vittorio, “Resonant second-harmonic generation in a GaAs photonic crystal waveguide,” Phys. Rev. B 68, 161306(R) (2003).
[CrossRef]

Weinreich, G.

P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118-119(1961).
[CrossRef]

Young, J. F.

A. R. Cowan and J. F. Young, “Mode matching for second-harmonic generation in photonic crystal waveguides,” Phys. Rev. B 65, 085106 (2002).
[CrossRef]

Appl. Phys. Lett. (1)

S. E. Harris, “Proposed backward wave oscillation in the infrared,” Appl. Phys. Lett. 9, 114 (1966).
[CrossRef]

IEEE Trans. Microw. Theory Technol. (1)

J. S. Savage and A. F. Peterson, “Higher-order vector finite elements for tetrahedral cells,” IEEE Trans. Microw. Theory Technol. 44, 874-879 (1996).
[CrossRef]

J. Opt. Soc. Am. B (2)

Nat. Mater. (1)

M. Soljacic and J. D. Joannopoulos, “Enhancement of nonlinear effects using photonic crystals,” Nat. Mater. 3, 211-219 (2004).
[CrossRef] [PubMed]

Opt. Express (1)

Opt. Quantum Electron. (1)

B. Maes, P. Bienstman, R. Baets, B. Hu, P. Sewell, and T. Benson, “Modeling comparison of second-harmonic generation in high-index-contrast devices,” Opt. Quantum Electron. 40, 13-22(2008).
[CrossRef]

Phys. Rev. A (1)

R. Iliew, C. Etrich, T. Pertsch, F. Lederer, and Y. S. Kivshar, “Huge enhancement of backward second-harmonic generation with slow light in photonic crystals,” Phys. Rev. A 81, 023820(2010).
[CrossRef]

Phys. Rev. B (7)

Y. Dumeige, F. Raineri, A. Levenson, and X. Letartre, “Second-harmonic generation in one-dimensional photonic edge waveguides,” Phys. Rev. B 68, 066617 (2003).
[CrossRef]

A. R. Cowan and J. F. Young, “Mode matching for second-harmonic generation in photonic crystal waveguides,” Phys. Rev. B 65, 085106 (2002).
[CrossRef]

A. M. Malvezzi, G. Vecchi, M. Patrini, G. Guizzetti, L. C. Andreani, F. Romanato, L. Businaro, E. Di Fabrizio, A. Passaseo, and M. D. Vittorio, “Resonant second-harmonic generation in a GaAs photonic crystal waveguide,” Phys. Rev. B 68, 161306(R) (2003).
[CrossRef]

J. Torres, D. Coquillat, R. Legros, J. P. Lascaray, F. Teppe, D. Scalbert, D. Peyrade, Y. Chen, O. Briot, M. Le Vassor d'Yerville, E. Centeno, D. Cassagne, and J. P. Albert, “Giant second-harmonic generation in a one-dimensional GaN photonic crystal,” Phys. Rev. B 69, 085105 (2004).
[CrossRef]

D. C. Marinica, A. G. Borisov, and S. V. Shabanov, “Second harmonic generation from arrays of subwavelength cylinders,” Phys. Rev. B 76, 085311 (2007).
[CrossRef]

K. B. Crozier, V. Lousse, O. Kilic, S. Kim, S. Fan, and O. Solgaard, “Air-bridged photonic crystal slabs at visible and near-infrared wavelengths,” Phys. Rev. B 73, 115126 (2006).
[CrossRef]

D. E. Aspnes and A. A. Studna, “Dielectric functions and optical parameters of Si, Ge, GaP, GaAs, GaSb, InP, InAs, and InSb from 1.5 to 6.0 eV,” Phys. Rev. B 27, 985-1009 (1983).
[CrossRef]

Phys. Rev. E (3)

M. Liscidini and L. C. Andreani, “Second-harmonic generation in doubly resonant microcavities with periodic dielectric mirrors,” Phys. Rev. E 73, 016613 (2006).
[CrossRef]

M. Luo and Q. H. Liu, “Spectral element method for band structures of three-dimensional anisotropic photonic crystals,” Phys. Rev. E 80, 056702 (2009).
[CrossRef]

M. Centini, C. Sibilia, M. Scalora, G. D'Aguanno, M. Bertolotti, M. J. Bloemer, C. M. Bowden, and I. Nefedov, “Dispersive properties of finite, one-dimensional photonic band gap structures: applications to nonlinear quadratic interactions,” Phys. Rev. E 60, 4891-4898 (1999).
[CrossRef]

Phys. Rev. Lett. (2)

Y. Dumeige, I. Sagnes, P. Monnier, P. Vidakovic, I. Abram, C. Mriadec, and A. Levenson, “Phase-matched frequency doubling at photonic band edges: efficiency scaling as the fifth power of the length,” Phys. Rev. Lett. 89, 043901 (2002).
[CrossRef] [PubMed]

P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118-119(1961).
[CrossRef]

Other (1)

R. L. Sutherland, Handbook of Nonlinear Optics, 2nd ed.(Marcel Dekker, 2003).
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

Spatial structure of two-by-two periods of an air-bridge PC slab with a square lattice of air holes in a slab with thickness h. The period of the square lattice is L x = L y , and the diameter of the air holes is D.

Fig. 2
Fig. 2

Band structure of the air-bridge PC slab with geometry shown in Fig. 1, and h = 0.439 L x , D = 0.416 L x , refractive index of the slab material n = 2.282 . The result of FDTD [19] is shown by the blue stars, and the result of the SEM is shown by the red dots.

Fig. 3
Fig. 3

Band structure of an air-bridge PC slab is plotted versus wavelength. The simulated PC slab has the geometry shown in Fig. 1 with L x = L y = 452 nm , h = 196 nm , and D = 190 nm . The dielectric material is GaP with the refractive index given in Ref. [20]. The nonlinear optical coefficients are neglected. The dotted line is the light cone in vacuum. The solid and dashed lines are the light lines with k x = sin ( 10 ° ) k 0 and k x = sin ( 45 ° ) k 0 , respectively. Each band is marked with an index, with the corresponding property indicated in Table 1. The bands with solid points are even mode (TE-like), and the bands with empty points are odd mode (TM-like). At the Γ point, the bands with red diamond points have a finite decay time and the bands with blue circle points have an infinite decay time.

Fig. 4
Fig. 4

Band structure of the ratio between the decay time and the corresponding period of each state. The index on each band corresponds to the index in Fig. 3. At the Γ point, the bands with blue circle points have an infinitely large decay time, so the ratios do not show up in the figure, and the bands with red diamond points have a finite decay time.

Fig. 5
Fig. 5

SHG exiting power rate (a) and stored energy (b) versus the SH wavelength for the system in Fig. 3 with nonzero nonlinear optical coefficients. The FF field is normal incidence. The resonant points of the light line and the band structure with finite (infinite) decay time at the Γ point give the SHG frequencies with an enhanced exiting power rate (stored energy).

Fig. 6
Fig. 6

SHG exiting power rate and stored energy versus SHG wavelength for the system in Fig. 3 with nonzero nonlinear optical coefficients. Forward and backward SH power rate when the incidence angle of the FF field is (a)  10 ° and (b)  45 ° . Stored SH energy for (c)  10 ° and (b) 45 ° incidence angle. The resonant points of the light line and the band structure give the SHG frequencies with an enhanced SHG effect.

Fig. 7
Fig. 7

Comparison of SHG exiting power rate for forward and backward directions in the air-bridge PC and the homogeneous slab when the incidence angle of the FF field is 45 ° . (a) The SHG exiting power rate for the forward and backward SH wave from the air-bridge PC slab and from the homogeneous slab. (b) The ratio between the forward and backward SHG exiting power of the air-bridge PC slab and the homogeneous slab.

Tables (1)

Tables Icon

Table 1 Symmetry Properties and Prominence in SHG Peaks of Each Band in Fig. 3

Equations (21)

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E = k E k Φ k ,
H ¯ = k up ( down ) η 0 H k Φ k ,
Ω [ ( × Φ i ) · μ r 1 · ( × E ) k 0 2 Φ i · ε r · E ] d V j k 0 Ω ( Φ i ) · ( n ^ × H ¯ ) d S = 0 ,
Ω [ ( × Φ i ) · μ r 1 · ( × E ) k 0 2 Φ i · ε r · E ] d V j k 0 Ω ( n ^ × Φ i ) · [ 1 2 H ¯ + L ( n ^ × E ) + K ˜ ( n ^ × H ¯ ) ] d S = j k 0 Ω ( n ^ × Φ i ) · H ¯ inc ,
j k 0 Ω ( n ^ × Φ i ) · [ 1 2 E + L ( n ^ × H ¯ ) K ˜ ( n ^ × E ) ] d S = j k 0 Ω ( n ^ × Φ i ) · E inc ,
L ( X ) = j k 0 Ω [ X ( r ) g P ( r , r ) + 1 k 0 2 · X ( r ) g P ( r , r ) ] d S ,
K ( X ) = Ω X ( r ) × g P ( r , r ) d S ,
g P ( r , r ) = K e j k 0 | r r K | 4 π | r r K | ,
S I I S I S S S I S S S + U M ( Re [ k 0 ] ) V M ( Re [ k 0 ] ) [ U E ( Re [ k 0 ] ) ] 1 V E ( Re [ k 0 ] ) E I E S = k 0 2 M I I M I S M S I M S S E I E S ,
S i , k = Ω d V ( × Φ i ) · μ r 1 · ( × Φ k ) ,
M i , k = Ω d V Φ i · ε r · Φ k .
U i , k M = j Re [ k 0 ] Ω ( n ^ × Φ i ) · L ( n ^ × Φ k ) d S ,
V i , k M = j Re [ k 0 ] Ω ( n ^ × Φ i ) · [ 1 2 Φ k + K ˜ ( n ^ × Φ k ) ] d S ,
U i , k E = j Re [ k 0 ] Ω ( n ^ × Φ i ) · L ( n ^ × Φ k ) d S ,
V i , k E = j Re [ k 0 ] Ω ( n ^ × Φ i ) · [ 1 2 Φ k + K ˜ ( n ^ × Φ k ) ] d S ,
S I I k 0 2 M I I S I S k 0 2 M I S 0 S S I k 0 2 M S I S S S k 0 2 M S S + U M ( k 0 ) V M ( k 0 ) 0 V E ( k 0 ) U E ( k 0 ) E I ( k 0 ) E S ( k 0 ) H S ( k 0 ) = P SHG ( k 0 ) H inc E inc ,
S I I 4 k 0 2 M I I S I S 4 k 0 2 M I S 0 S S I 4 k 0 2 M S I S S S 4 k 0 2 M S S + U M ( 2 k 0 ) V M ( 2 k 0 ) 0 V E ( 2 k 0 ) U E ( 2 k 0 ) E I ( 2 k 0 ) E S ( 2 k 0 ) H S ( 2 k 0 ) = P SHG ( 2 k 0 ) 0 0 ,
P i SHG ( 2 k 0 ) = Ω d V Φ i · P ( 2 k 0 ) .
P i SHG ( k 0 ) = Ω d V Φ i · P ( k 0 ) .
R Forward , Backward = Bottom , Top ( z ^ ) · Re ( 1 2 E × H * ) d S Top ( z ^ ) · Re [ 1 2 E inc × ( H inc ) * ] d S .
W e = Ω 1 2 E * · Re [ ε r ] · E d V .

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